The present invention relates to a computerized simulation method suitable for fluid analysis for instance, more particularly to a method which can solve a nonuniform Poisson equation.
In a computerized simulation to analyze a fluid flowing in a given three-dimensional space, a mesh of the three-dimensional space is usually defined by utilizing a Cartesian grid, and at each grid point, the pressure, temperature and/or flow velocity of the fluid are calculated, wherein if the fluid can be considered as incompressible, the equation regarding the pressure is usually given as a Poisson equation. The Cartesian grid can be a rectangular Cartesian, cylindrical Cartesian and spherical Cartesian.
As a method to solve a Poisson equation, iterative methods such as Jacobi method, Gauss-Seidel method and SOR method have been known.
In a iterative method, usually, initial values are given firstly (step 1), and a computation is carried out (step 2), then a convergence test is carried out to determine whether or not the solution satisfies a judgment condition, and if the judgment condition is not satisfied, the step 2 is repeated until the solution satisfies the judgment condition (step 3).
In such a iterative method, when the size of the matrix to be calculated is increased, the solution becomes poorly-converged, and thereby a long computational time is required.
In recent years, therefore, an Algebraic Multigrid (AMG) Method is widely used. As well known in the art, the AMG method is a method extended so as to solve an unstructured grid in addition to an orthogonal grid, on the basis of the geometric multigrid method. Like the original multigrid method, the extended AMG method has a characteristic that the number of repetitions is independent of the mesh size.
Currently, it is considered as the most high-speed solver for a simultaneous linear equation.
Thus, the AMG method has been widely employed in the commercially offered general-purpose solvers (e.g. “FLUENT” of Ansys, Inc, “STAR-CD” of CD-Adapco).
As a solver for a large-scale matrix, the AMG method is excel in the convergence, but a large memory size is required for the computer.
on the other hand, a Block-cyclic Reduction Algorithm is known as a direct method in which a large-scale matrix can be solved without requiring a large memory size and the convergence is to machine precision.
However, the Block-Cyclic Reduction Algorithm is premised on that the constant “k” in a Poisson equation (for example, the “k” in the expression (1) in claim 1) is a fixed value. If the values of the “k” is not a fixed value and accordingly it is impossible to put the “k” outside the Poisson matrix as a common coefficient, then the Block-cyclic Reduction Algorithm can not be used to obtain the solution directly.
In order to use a Block-cyclic Reduction Algorithm properly, the values of the “k” used in a simulation must be a fixed value. In other words, it is necessary to use an explicit method (to satisfy the courant-Friedrichs-Lewy (CFL) condition) and to decrease the time intervals (steps) of the computing. Therefore, there is a disadvantage such that the computational cost increases.
The above-mentioned CFL condition is a necessary condition such that the velocity of propagation of data during computing in a simulation or numerical analysis must be faster than the velocity of propagation of wave or physical value in the real phenomenon to be simulated.
For instance, in the case that a wave motion in a discrete lattice system should be dealt with in the simulation, the values of the time intervals (at) used in solving of the motion equation for the numerical solution must be smaller than the time required for the real wave motion to progress between the adjacent grid units or cells.If the value of Δt exceeds the upper limit determined by the CFL condition, then numerical divergence occurs, and it becomes impossible to obtain a sensible solution.If the grid spacings are decreased in order to make a detailed simulation or analysis, the upper limit for the time step also decreases.
In the case of an advection equation, the CFL condition is given asΔx/Δt>C whereinΔx is a grid spacing,Δt is the computational time interval, andC is the velocity of real wave (or characteristic velocity).
This CFL condition becomes a condition to be used when a temporal progress is made in an explicit method.
In order to avoid the use of such condition, an implicit method is often employed.
one such popular method is SIMPLE (Semi Implicit Method for Pressure Linked Equations) method which is offered by commercial softwares like FLUENT of Ansys Inc. or by StarCCM+ of CD-Adapco. This method requires sub-iterations and all the Navier Stokes variables like velocity field and pressure field are corrected and updated to latest values for current time level. Since the gradients of velocity field required to update solution to next level are no longer the values of previous time level, the solution is essentially an implicit type method. For this reason, solution is numerically stable even for very large values of time step sizes.